Divergence Definition and 746 Threads

[SOLVED] Divergence, nabla Homework Statement Given the vector, find the dot product. Homework Equations dot product of nabla and the vector is just partial derivative of each component. The Attempt at a Solution I'm trying to figure out if I can just leave out the...

Homework Statement Investigate the behavior (convergence or divergence) of \sum_n 1/(1+z^n) where z is complex. Homework Equations The Attempt at a Solution If the modulus of z is less than 1, it is not hard to show that the limit of the sequence is not 0 (it is actually not finite) and thus...

I've been reading up on these three recently, and wondered if anyone could confirm what I think they do. I'm not 100% I understand these. del (\bigtriangleup), when applied to a scalar, creates a vector with that scalar as each of the XYZ values. eg \bigtriangleup . x = (x,x,x)...

I have a few series which I'm having trouble proving whether they converge or diverge. I know the following tests for convergence: comparison test, ratio test, n-th term test, and root test. Here are the series and what I have tried so far: \sum n -1 / n2 : I'm assuming this series diverges...

Homework Statement Hi all. Please take a look at the following problem: Evaluate the surface integral \int{F \cdotp d\vec{S}} for the following vector field: F(x;y;z) = xyi + yzj + zxk, where i, j and k are unit vectors. S is the part of the paraboloid z = 4-x^2-y^2 that lies above the square...

Can anyone tell me whether or not the divergence theorem requires a conservative vector field? On a practice exam my professor gave a vector field that was nonconservative (I checked the curl) and proceeded to perform the divergence theorem to find the flux. On one of my homework problems I...

I want to calculate the divergence of a two dimensional 3 order tensor; e.g. nabla=(d/dx, d/dy) and Ax = ( C D) ( E F), Ay = ( G H) ( I J) (it's a 2x2x2 cube). Index notation: (nabla)_i = d/dx_i and elements of A are A_ijk How do I contract it properly...

[SOLVED] Beam Divergence Homework Statement A 1.5mW helium-neon laser beam delivers a spot of light 5mm in diameter across a room 15m wide. The beam radiates from a small circular area of diameter 0.5mm at the output mirror of the laser. Assume that the beam irradiance is constant across...

S\int\int F*Nds F(x,y,z) = (xy^2 + cosz)i + (x^2*y + sinz)j + e^(z)*k s: z = 1/2\sqrt{x^2 + y^2} , z = 8 divF = y^2 + x^2 +e^z Q\int\int\int (y^2 + x^2 + e^k)dV This is as far as I got, I have no idea how to do the limits for this triple integral thanks in advance guys.

When do I use the Test for Divergence. I am confused because on some problems I get that the limit of the equation is not equal to 0 and it is convergent. But using the Test for Divergence every answer I had in the other problems would be contrary to the answer I got which I know to be right. I...

Hello - I'm supposed to derive the divergence formula for spherical coordinates by carrying out the surface integrals of the surface of the volume in the figure (the figure is a piece of a sphere similar to a box but with curves). The radial coord is r. The polar angle is \varphi and the...

Homework Statement Determine whether the following integral is convergent or divergent. If convergent, what does it converge to? dx/(4x^2 + 4x + 5) [-infinity, infinity] Homework Equations comparison theorem?The Attempt at a Solution I think it is convergent, so I set the original integral...

Consider a cylindrical shell so that the cross sectional radius is some constant a. In the first term of the divergence expression in cylindrical coordinates: \frac{1}{r}\frac{\partial}{\partial r}(rA_{r}) When I multiply the radial component by r, do I go ahead and substitute r=a...

Hello everyone, I'm new to this forum. I have a doubt about differential forms, related to the divergence. On a website I read this: "In general, it is true that in R^3 the operation of d on a differential 0-form gives the gradient of that differential 0-form, that on a differential 1-form...

Homework Statement Can someone please explain to me what the physical meaning of the divergence integrals and curl integral is? In the problems I have come across, they ask us to calculate areas and etc.. using Green's theorem. Which one should I use in that case? Thank-you very much for...

Suppose that a_n\geq 0 and there is \lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=c If c>1,series diverges. if c<1 series converges. For a_n=\frac{n!}{n^n} \lim_{n\rightarrow\infty}\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n} \lim_{n\rightarrow\infty}\frac{n^n}{(n+1)^n} Then I used...

The sum of 1-2+3-4+5..., and divergence The sum of 1-2+3-4+5..., which can be written as Diverges for m = infinity, yet there are postulates that this is equal to \frac{1}{4}. First, I don't understand how you can obtain a fraction out of a natural numbers if they are consecutively...

Homework Statement Show divergence theorem works For the vector field E = \hat{r}10e^{-r}-\hat{z}3z Homework Equations \int_{v}\nabla \cdot E dv = \oint_{s} E \cdot ds The Attempt at a Solution \nabla \cdot E = 1/r \frac{d}{dr}(rAr)+1/r\frac{dA\phi}{d\phi}+\frac{dAz}{dz}...

I am trying to understand the derivation of the divergence formula in cylindrical coordinates. www.csupomona.edu/~ajm/materials/delcyl.pdf paper does a good job of explaining it but I don't understand 2 things that the author does. \frac{\partial\hat{\rho}}{\partial\phi} = \hat{\phi} and...

Homework Statement Determine whether the following sequence, whose nth term is given, converges or diverges. Find the limit of each convergent one. n[1 - cos(2/n)] Homework Equations I have made a solid attempt and obtained an answer but I am convinced I made a mistake and have missed...

i having a some trouble verifying the Divergence theorem for A=y^2zex-2x^3yey+xyz^2ez with respect to V being a unit cube

The fundamental theorem of calculus is basically the divergence theorem but dealing with a ball in R^1 instead of a ball in R^3. The fundamental theorem of Calculus relates the stuff inside the ball to its boundary, just like how the divergence theorem relates the stuff inside a volume with its...

Are there versions of the divergence theorem that don't require a compact domain? In my reference literature the divergence theorem is proved under the assumption that the domain is compact.

Been working my way thru H.M. Schey-been out of college for 50 yrs. This problem has me stumped. F (of x,y,z)= i (f of x) + j (f of y)+k f (-2z). F is a vector function, and i,j,k are unit vectors for x,y,z axis. The problem is to find Div F., and then show it is 0 for the point c,c...

I am trying to think of a non-constant function whose divergence and curl is 0. It seems like this is impossible to me. Any hints?

divergence question show that the divergence transforms as a vector under 2D rotations. I am so confused abouth what this question wants me to do. Obviously the divergence is not invariant under rotations. Consider the divergence of the function f(x,y) = x^2 * x-hat. The divergence is...

My book says that divergence can be understood in the context of fluid flow as the rate at which density flows out of a given region. It says that if F(x,y,z) is the velocity of a fluid, then that is the interpretation of the divergence. I fail to understand where the density comes in when we...

Hello; I remember the days of muti variable calculus. The man said that divergence is equal to del dot the vector field. So on the exam he gave us a vector field, and I did del dot the given vector field and won big time. The other day I decided my concentration would be...

Homework Statement I'm supposed to evaluate the following series or show if it diverges: SUM (sigma) log [(x+1)/x] Homework Equations Drawing a blank...:confused: The Attempt at a Solution I'm unsure how to start this. We've gone over all sorts of tests for convergence (ratio, comparison...

We know that div \; (\hat{r} / r ) = 4 \pi \delta (r) Why is there no generalized function (distribution) for div \; (\hat{r} / r^2) = ??

Homework Statement n / 8^nHomework Equations The Attempt at a Solution It converges to 8 / 49? Not sure how. First test of divergence lim n-> n / 8^n. infinity / infinity = 1. BUT bottom grows fast. Using L`Hospital lim n -> 1 / 3*8^n*ln(2) ---> goes to 0 Tried to use the ratio test [ 1 /...

How can i do the divergence of a matrix 3x3?

Question Evaluate both sides of the divergence theorem for V =(x)i +(y)j over a circle of radius R Correct answer The answer should be 2(pi)(R^2) My Answer the divergence theorem is **integral** (V . n ) d(sigma) = **double intergral** DivV d(tau) in 2D. Where (sigma)...

Why does an enclosed point charge have zero divergence/flux? Mathematically I can see that when the divergence operator is applied to E=q/r^2 (pointing in the r direction) I get zero, but what is the physical explanation for what is going on? I am confused because other enclosed electric fields...

Let us consider a collection of non-interacting hydrogen atoms at a certain temperature T. The energy levels of the hydrogen atom and their degeneracy are: En = -R/n² gn = n² The partition function in statistical physics is given by: Z = Sum(gn Exp(-En/kT), n=1 to Inf) This...

Homework Statement Does this series converge or diverge? Series from n=1 to infinity n(-3)^(n+1) / 4^(n-1) Homework Equations The Attempt at a Solution Okay, I've tried it both ways. Ratio test: lim n --> inf. ((n+1)*(-3)^(n+1)/4^n) / (n * (-3)^n / 4^(n-1)) Now...

Homework Statement Does the sum of the series from n=1 to infinity of 1+sin(n)/10^n converge or diverge. Homework Equations The Attempt at a Solution I can use the comparison test or the limit comparison test. I'm not sure where to go from here.

Homework Statement the question is: Without doing any calculations, explain why the divergence of each of the functions must be zero(Hint: consider what electric field these functions physically correspond to) P.S: I make the unit vectors in bold. A1=x A2=ρ(1/ρ) with ρ>0...

Homework Statement Find \int_{s} \vec{A} \cdot d\vec{a} given \vec{A} = ( x\hat{i} + y\hat{j} + z\hat{k} ) ( x^2 + y^2 + z^2 ) and the surface S is defined by the sphere R^2 = x^2 + y^2 + z^2 directly and by Gauss's theorem. Homework Equations \int_{s} \vec{A} \cdot d\vec{a} =...

I am studying divergence and curl in my E&M class. I was wondering, why is divergence a useful concept? I mean, for point charges, the divergence is zero everywhere except where the charge is located. Even for charged surfaces, \nabla\cdot E = \frac{\rho}{\epsilon} Loooking at this it seems...

Homework Statement If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector? Homework Equations Just the equations for divergence and curl. In TeX: \nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial...

Homework Statement Prove the divergence of the harmonic series by contridiction Homework Equations Attached file The Attempt at a Solution I understand what they are doing in the first two lines, however, the lines after assuming the series converges with sum S, confuses me. They...

Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge...

Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge...

Hi, I hope this is advanced enough to warrant being in this section: I'm supposed to use the Gauss theorem (and presumably his law) to show: 1)The charge on a conductor is on the surface. 2)A closed hollow conductor shields its interior from fields due to charges outside, but doesn't...

a_n = (1 + k/n)^n Determine the converge or divergence of the sequence. If it is convergent, find its limit. My professor said to convert the sequence to f(x) and use ln (ln y) and L'Hospital's Rule. Do I have to use ln? Is there another way to find the convergence?

I am going through Boas.Ch-1.on infinite series. Can anyone help? 1.May we use preliminary divergence test for series with +ve and -ve terms?How?For some situation occurs when we are supposed to make out (-1)^infinity

[SIZE="5"]The Background: I'm trying to construct a rigorous proof for the divergence theorem, but I'm far from my goal. So far, I have constructed a basic proof, but it is filled with errors, assumptions, non-rigorousness, etc. I want to make it rigorous; in so doing, I will learn how to...

Prove that if a_{n} > 0 and \sum a_{n} converges, then \sum a_{n}^{2} also converges. So if \sum a_{n} converges, this means that \lim_{n\rightarrow \infty} a_{n} = 0 . Ok, so from this part how do I get to this step: there exists an N such that | a_{n} - 0 | < 1 for all n > N...

is (DEL dot A) the same as (A dot DEL)? I know the dot product is commutative, but this involves an operator. if the answer is YES, than why does one of the product rules read like this: DEL X (A X B) = (B dot DELL)A - (A dot DEL)B + A(DEL dot B) - B(DEL dot A) they commute the...